Tagged Questions

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

0
votes
1answer
13 views

Conditional Probability - chance for an event to happen

I am learning probabilities at the moment and I have come across this problem: A person takes four tests in succession. The probability of his passing the first test is p, that of his passing each ...
2
votes
1answer
20 views

Expectation and Variance of stochastic equation

My questions is related to this question: Stochastic Differential equation, expectation and variance I.e how do you calculate the variance and expectation of $U_t = e^{-\gamma t}U_0 + \int_0^t ...
0
votes
0answers
18 views

Gaussian random processes

Let A B and C be Gaussian processes. If A&B are jointly Gaussian, and B&C are jointly Gaussian then A and C are jointly Gaussian? Is this statement true? and can it proven? (or any ...
0
votes
1answer
30 views

The ito integral is gaussian [duplicate]

Let $\Omega, F, P)$ be the classic setting. I saw that if $f$ is a function which satisfies some assumptions then the integral with respect to the brownian motion is Gaussian. Ie $\int_{0}^{t} f_u ...
0
votes
0answers
4 views

Gaussian conditional distribution

Let $Y_t$ be a gaussian process with $E[Y_t]=0$ and $Z=\frac{\int_0^1 Y_s ds}{\sqrt{V}}$ where $V=Var(\int_0^1 Y_s ds)$ (so Z has a standard normal distribution). I want to prove that conditionally on ...
2
votes
1answer
79 views

Ito's Lemma for negative exponential

I'd been reading on Hull-White model, when I encountered the bond-pricing formula, that is if $$ dr(t) = (\alpha(t)-\beta(t)r(t))dt + \sigma(t)dW(t)$$ for some deterministic function $\alpha, \beta, ...
2
votes
0answers
26 views

Layman perspective of mean time spent in transient state of a Markov chain.

Let $X=\{X_n\}$ be a finite state Markov Chain with the state space $\{0,1,2,\ldots,N\}$ such that $0$ is the single absorbing state and all the rest states are transient. The following is the ...
0
votes
1answer
13 views

What is the average of two stochastic processes multiplied?

Consider two random processes $X(t)$ and $Y(t)$ for which $$\langle X(t) X(t') \rangle = \mu_X^2 + \sigma_X^2 \delta(t-t')$$ $$\langle Y(t) Y(t') \rangle = \mu_Y^2 + \sigma_Y^2 \delta(t-t')$$ ie. the ...
1
vote
1answer
15 views

wide sense stationary process

I am a beginner in probability theory, and have a very basic and limited understanding of the random processes, but interested in understanding them more. In case of wide sense stationary ...
1
vote
4answers
64 views

Find the expected number of steps needed until every point has been visited at least once.

The complete graph on {1,...,N} is the simple graph with these vertices such that any pair of distinct points is adjacent. Let $X_{n}$ denote the simple random walk on this graph and let T be the ...
4
votes
0answers
56 views

Converge to Brownian Motion problem

Consider the following sequence of SDEs: $dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$ Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion. I have been ...
0
votes
0answers
24 views

Stochastic processes

Update I am a bit confused whether $y_t$ is independent over time under the following assumptions: Consider, first a RV $A$, that follows this process: $A_t = \rho A_{t-1} + e_t$, where $e_t$ is ...
2
votes
1answer
32 views

Proof of the Début theorem

I was reading the stochastic calculus notes on this website and I read the following in the proof of the Début theorem but I could not understand what does it mean. Can someone explain it to me ...
1
vote
0answers
8 views

Time changes conditions to be adapted

Given a process $X_t$ and another process $T_t$ which is increasing, what conditions should we require such that the process $X_t$ is adapted to the time change $T_t$, that is such that $X_t$ is ...
4
votes
1answer
76 views

Sum of Brownian Motions

I've got a little problem: if $X_{t}$ and $Y_{t}$ are two indipendent Brownian motions, is then $$Z_{t}:=X_{t}+Y_{t}$$ a Brownian motion too? I've got some troubles only with showing that $Z_t$ is ...
1
vote
1answer
36 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
1
vote
1answer
21 views

All right or left continuous processes are jointly measurable

I was reading the Stochastic Calculus notes by George on this website http://almostsure.wordpress.com/2009/11/03/stochastic-processes-indistinguishability-and-modifications/ and I cannot understand ...
0
votes
1answer
28 views

Can someone help me understand the following?

I was reading George Lowthers notes on Stochastic Calculus and , he says the following but I cannot figure out what it exactly means? ...
1
vote
1answer
14 views

Existence of Truncated Brownian motion?

I am currently engaging a research that would really use your help. I am considering add a brownian-type shock to a "fraction" $\theta \in [0,1]$, for example $$d\theta_{t} = \sigma \theta_{t} ...
3
votes
3answers
78 views

1-dimentional stochastic differential equation

I would like to solve this SDE $$dX_{t}=\left(\sqrt{1+X^{2}}+\dfrac{1}{2}\right)dt+\sqrt{1+X^{2}} dB_{t}$$ I've tried to solve first the homogeneous equation ...
1
vote
0answers
29 views

Measurability properties of processes that arise as limits of sequences of measurable processes

I try to reduce my problem to a more general statement from which I want to know whether this is true in general. I have a sequence of continuous-time stochastic processes $X_t^{(n)}, t \geq 0$ with ...
0
votes
1answer
25 views

Nonhomogeneous Poisson process

Let $\lambda:[0,\infty)\rightarrow [0,\infty)$ be a continuous function and $N$ be a Poisson process with rate $1$. Define $\Lambda(t)=\int_0^t{\lambda(x)dx}$ then how do we prove that $$ ...
0
votes
1answer
20 views

Proof of the progressiveness of a stopped progressive process

I have trouble understanding the proof of Proposition 2.18 in Karatzas/Shreve: Brownian Motion and Stochastic Calculus, which states that a $\mathcal{F_t}$-progressively measurable process $X_t$, ...
3
votes
0answers
25 views

Cubature on Wiener space

Suppose $(X_t)_{t\geq 0}$ diffuses as, $$ dX_t = \mu(X_t)\, dt + \sigma(X_t) \, dW_t $$ and, $$ g(t,x)=\mathbb{E}[g(T,X_T)\vert\mathcal{F}_t] $$ By Feynman-Kac we have, $$ ...
4
votes
2answers
69 views

Uniform distribution, as a sum of biased Bernoulli trials.

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
0
votes
0answers
26 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
1
vote
2answers
18 views

$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ ...
0
votes
1answer
27 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let ...
0
votes
0answers
17 views

Comparing frequencies in stationary distribution

Do there exist theorems for comparing frequencies in the stationary distribution of a (say) aperiodic, positive recurrent Markov chain? i.e. given the transition probability matrix $\mathbf{P}$ with ...
0
votes
1answer
17 views

Transition matrix to graph

Is there a program which can given a transition matrix $P$ draw a graph from a it? The transition matrix is also known as stochastic matrix and probability matrix see ...
1
vote
1answer
12 views

proof T is a stopping time

let $X(t) $ is a stochastic process and is cadlag and adapted, let $T = \inf\{t:|X(t)| \ge c\}$, proof T is a stopping time. i.e.$\{T\le t\} \in F_t$
2
votes
1answer
45 views

Expected Service Times for truncated exponential

I'm trying to solve a problem where all arriving items (arrival exponential $\lambda = 1/5$) are divided into into groups, those who are served within 5 units of time and those who have their service ...
1
vote
0answers
19 views

Is this application of the law of total probability correct?

Let us consider a counting process $N(t)_{t\geq0}$ which is neither Markovian nor Levy. Is it correct to write $$ \mathbb{P}(N(t)=j)=\int_{0}^{t}\mathbb{P}(N(t)=j, N(s)=i)ds $$ for $j\geq 1$ and ...
1
vote
0answers
35 views

Convergence of the sum of two stochastic processes

I've one question regarding the convergence of the sum of two stochastic processes. Let $(X^n_{t})_t \rightarrow (X_t)_t$ and $(Y^n_{t})_t \rightarrow (Y_t)_t$ for $n \to \infty$ where $\rightarrow$ ...
0
votes
0answers
7 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
1
vote
1answer
29 views

Expectation over 2 random variables, help needed

Hi I am new here and I hope I can get some help. My question is about taking expectation over random variables. Lets say I have two random variables $\Xi$ and $\theta$ where $\Xi$ is for example a ...
1
vote
1answer
30 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
1
vote
2answers
28 views

What is the expected number of flips that are needed?

Suppose we flip a fair coin repeatedly until we have flipped four consecutive heads. What is the expected number of flips that are needed? The hint is given is as follows: Consider a Markov chain ...
0
votes
1answer
21 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
2
votes
1answer
27 views

Question about the Poisson process

A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of ...
1
vote
0answers
25 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
0
votes
1answer
24 views

Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
0
votes
0answers
17 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
0
votes
0answers
19 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
2
votes
1answer
23 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
1
vote
0answers
15 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
1
vote
0answers
27 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
0
votes
0answers
29 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
0
votes
0answers
26 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
0
votes
0answers
21 views

Verifying solution of difference equation?

I have the following difference equation - $2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$ The boundary conditions are $h_0 = 1$ and $h_{20} = 0$ How would I go about verifying that $h_x = ...